Easy to diagnose an alternator with a digital voltmeter (start under $20 and a real necessity on any RV).
Engine off, shore power off. Check chassis battery voltage at the battery. Fully charged is 12.7 VDC. Surface charge may have it above that, though turning on headlights with all charging devices off for a few minutes will dissipate the surface charge.
Start engine and have someone hold it at high idle (1000 to 1200 RPM). Recheck voltage at the chassis battery. Should be around 14 VDC.
Turn on headlights, dash HVAC fan to high-- i.e. turn on some high amp loads. Voltage at the chassis battery should stay around 14 VDC.
Let us know what you find.
